I have owned three 10/22's and have had a lot of success with all of them. With factory stock and lite barrel, they don't shoot THAT well at 100 yds, but they may be able to get the job done. Having said that, i havent owned one of the customized 10/22's or factory 10/22T's with the bull barrell. From my understanding these guns should give the kind of accuracy your wanting.

I have no experience whatsoever with the 597, in fact i dont even know anybody who has one. The 10/22's are so popular around here that 90% of hunters and shooters use them. I know hunting rabbits your definetely leaning toward a semi, but if you have thought about a bolt .22, look at the CZ's. I have a cz in .17 hmr and it is the most accurate rimfire i have ever owned. I'll soon be buying a cz .22lr with a bull barrell. These are excellent guns that i believe will outshoot a ruger 77/22 or remington 504 any day of the week.